I have almost finished a book called 'the Music of the Primes" by Marcus de Satoy. It is mostly about the unsolved Riemann Hypothesis dealing with the critical line of zeros in the Zeta function. If you havent heard of it then check out the book, it is a very non mathematical account of the development of the hypothesis and is more of a historical account than a math textbook.
As some of you probably know, Riemann hypothesis is one of the Clay Millenium problems that has a million dollar price tag on it (sad that the longer it takes to solve the less the solution will be worth, and we have Ben to thank for that). In the spirit of this, I thought that I would propose a question about options for some of the younger monkeys to think about. You will not find the answer in Hull or Natenberg, although those books give you the tools to come up with an answer. There is also probably not just one solution, so get creative with your answers. If i see an answer that is creative altho wrong, I will reward it.
TASK:
-Describe a situation/vanilla options position where you are long gamma and receiving theta at the same time
I have 69 silver bananas to give away and the first person to get the answer I am thinking of will get a free copy of my trading interview guide.
All you pro's out there stay away, this is only open to people who are not on a derivatives desk. It is also not open to people who have bought my interview guide already as you probably know the answer.
I will post my solution at the end of the week/when my bananas run out.
Long call ATM, short puts OTM w/ puts being out a month from the calls. Depending on the strike you choose, the puts shouldnt eat away all your gamma since it will be high ATM, and the theta on the shorts being out a month shouldnt be completely offset by the ATM long.
Gamma of an OTM longer dated put > Gamma of an OTM shorter dated put (if far enough OTM as there is an inflexion point)
So you are saying you will be long gamma on the call that is larger than the gamma on the short put, and you are paying theta on the call that is larger than you receive on the put, that means you are long gamma and paying theta, although less in an absolute sense as the put offsets some of it.
If you are thinking along the lines that the OTM put will receive more theta because it is longer dated, you will also have to consider the fact that gamma for the OTM put will be higher than you would expect as well.
i was thinking that my multiple short puts being fwd dated would have enough long theta to offset the negative theta on the long call that is ATM. I was also thinking that my long call being ATM would have a larger gamma and offset the negative gamma on the short puts that are OTM. I figure the amount of contracts would have to be played with depending on what the underlying was, how volatile, etc, but in general i was selling more puts fwd than long calls spot. (spot and fwd being more illustrative and not referring to literal settlement).
I figured this would give you positive gamma and theta and cause you to lose on vega. If the gamma on the fwd short puts would be greater (more negative) than the calls, then yeah, this wouldn't work. I'm sure it has to be some type of butterfly/spread/strangle combo that gets you there though. i'll see if i can come up with something else. Thanks for the feedback.
I'll take a stab at this:
I believe the theta of a put option is always higher than the theta of a call at the same strike if we're using the standard equation for theta (ie less negative). And we also know that gamma of a put is equal to the gamma of a call at the same strike. So to create a position with positive gamma/theta, we can take a long position in a put that has slightly more gamma than a call, but that still has a less negative theta than the call, and we short the call. So an example of this would be a short risk reversal position (long a low strike put, short a slightly higher strike call). I believe it is possible to have positive theta/gamma with this position (I believe you can also get that with a long deep in-the-money put option as well, but I don't think that's as interesting).
What I would like to know however is what are the ramifications of this? ie why isn't this strategy more wildly popular given that its relatively less risky in terms of adverse greek positions?
Gamma and theta match up pretty well, so if one has higher theta, it will have higher gamma as there is a no arbitrage relationship between the two.
But you are on the right path, it is to do with risk reversals. Its not a static relationship that just exists though, its something to do with the dynamics of the risk reversal once the stock moves.
The ramifications arent huge, its really a technicality and isnt really exposable to a huge extent. More of an exercise to make people think a bit more creatively and in multiple dimensions.
I'm not sure I buy that no-arbitrage relationship argument. Let me try to lay it out again. So first, lets forget about the risk reversal and consider a position where you long a put and short a call at the same strike. Using my trusty Hull textbook, the equation for theta for a put and call are as follows:
Note that when we buy the put and short the call that ugly first term becomes irrelevant, as thetas are additive, so if we buy one unit of the put and short one unit of the call, they cancel out. Also note that
So that simplifies the theta(put) equation to:
So, buying the put and subtracting the call gets you:
Also note that, while I haven't computed the gamma, the gamma of such a spread will always be 0, as gamma(put) = gamma(call) at the same strike, and since gammas are additive, gamma(put) - gamma(call) = 0.
So, we can see that the theta of a long-put, short-call at the same strike spread always generates a positive theta, and is always flat gamma. So what I've proven is that it is possible to have a scenario with positive theta, flat gamma.
And while that doesn't directly prove there is a scenario with positive theta, positive gamma, it can be inferred that since there is a strike that exists with positive theta, flat gamma that there MUST be a strike that exists with positive theta, positive gamma, given an ever so slight adjustment to give the put slightly more gamma (ie move it slightly closer to ATM than the call) but not enough to offset that +rKe^(-rT) in theta.
I'm not sure I see any obvious flaws yet in my thinking, so please let me know if I have made any mistakes and where.
Best reason why I can think of that the no-arbitrage relationship between theta and gamma doesn't necessarily hold is because higher order terms for both greeks are ignored so the no-arbitrage equation should only be approximately but not exactly true? Not sure of this one, however.
My solution involves a basket of options (shorts and longs).
Modeling Vega and Gamma sensitivity as normally distributed about the Strike (not exactly true but we can live with that), the peak for Vega increases with Strike (K), and for Gamma decreases with strike.
Let's now create a portfolio of options inversely weighted by K^2. This should make your portfolio constant Vega, hypothetically, yes? You can drive this constant sensitivity line down to 0 by hedging the Vega with a linear Vega instrument like a Variance swap.
So you are flat vega across strikes, so you are immune to changes in implied vol, how does this make you long gamma and receive theta? You might have a solution but you need to explain that connection as Im not really seeing it.
Looking at the BS formulas, yes you might have an extremely tiny bit of theta as shown by the formulas, but are you really receiving theta? Because you are basically short the stock, so you can go long the stock and have a perfect hedge. If you open up a greeks position analyzer (if you dont have one ill send you mine), and put this position in, then you will see that there is 0 gamma and some very little theta, but to get this theta in trading P&L you need for the price of your spread to slowly rise (if you are long the spread and receiving theta), this doesnt happen however. Or it does but is so insignificant that it doesnt register. Therefore are you really receiving theta?
Ill still give an SB for that though.
Ill also post the no arb argument of gamma vs theta, its not a static argument, more through delta hedging.
You're right, the long put, short call at the same strike is essentially a synthetic stock position, with 0 greeks except theta.
What I'm ultimately suggesting though is a very very narrow (not sure if thats the right term?) risk reversal (ie long put at strike K and short call at K+ .1 or something). That SHOULD generate a tiny positive gamma position as well instead of 0 gamma as you say. I would love to try it out on the greeks position analyzer though. I'd really appreciate if you could PM it over! I agree, it will in all likelihood be a very very tiny amount of theta/gamma however, but still theoretically positive.
Also, if I'm not mistaken, I believe a sufficiently deep ITM put option by itself should create the positive gamma/positive theta position as well. So I do believe its possible to create it statically rather through a dynamic strategy (as it seems like you may be trying to hint at?)
Send me ur email in a PM and ill send u the excel file.
HINT 1: Think about a 95/105 (spot = 100) risk reversal (long 95 put, short 105 call) with a negative skew
(by negative I mean upside strikes have higher implied vol than downside ones, I know some people might say negative means negative slope, but its easier to think of this way because the general state of the skew is that downside strikes are more expensive and so you think of skew increasing if downside strikes get more expensive)
long downside strikes trading at a low vol, short upside strikes at a high vol. Same expiry, higher vol option has a greater decay than the option you are long. Also low vol option has higher gamma than higher vol option.
For OTM/ITM options gamma is higher for a higher vol option. Therefore your case could work but you would need to have strikes whithin the critical strikes where higher vol leads to higher gamma. +1
The case I was thinking of deals with this strategy but in the special case of a negative skew and negative correlation between spot and implied volatility. Then look at happens to delta as the stock moves either way.
derivstrading, I know this may not be creative, but you are long gamma, long theta for Deep ITM European puts. Was this something along the lines you were looking for?
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